diff --git a/openmp/libomptarget/src/omptarget.cpp b/openmp/libomptarget/src/omptarget.cpp
--- a/openmp/libomptarget/src/omptarget.cpp
+++ b/openmp/libomptarget/src/omptarget.cpp
@@ -545,6 +545,13 @@
       DP("Mapping does not exist (%s)\n",
          (HasPresentModifier ? "'present' map type modifier" : "ignored"));
       if (HasPresentModifier) {
+        // OpenMP 5.1, sec. 2.21.7.1 "map Clause", p. 350 L10-13:
+        // "If a map clause appears on a target, target data, target enter data
+        // or target exit data construct with a present map-type-modifier then
+        // on entry to the region if the corresponding list item does not appear
+        // in the device data environment then an error occurs and the program
+        // terminates."
+        //
         // This should be an error upon entering an "omp target exit data".  It
         // should not be an error upon exiting an "omp target data" or "omp
         // target".  For "omp target data", Clang thus doesn't include present
@@ -564,6 +571,14 @@
          DataSize, DPxPTR(TgtPtrBegin), (IsLast ? "" : " not"));
     }
 
+    // OpenMP 5.1, sec. 2.21.7.1 "map Clause", p. 351 L14-16:
+    // "If the map clause appears on a target, target data, or target exit data
+    // construct and a corresponding list item of the original list item is not
+    // present in the device data environment on exit from the region then the
+    // list item is ignored."
+    if (!TgtPtrBegin)
+      continue;
+
     bool DelEntry = IsLast || ForceDelete;
 
     if ((ArgTypes[I] & OMP_TGT_MAPTYPE_MEMBER_OF) &&
diff --git a/openmp/libomptarget/test/mapping/data_absent_at_exit.c b/openmp/libomptarget/test/mapping/data_absent_at_exit.c
new file mode 100644
--- /dev/null
+++ b/openmp/libomptarget/test/mapping/data_absent_at_exit.c
@@ -0,0 +1,33 @@
+// RUN: %libomptarget-compile-run-and-check-aarch64-unknown-linux-gnu
+// RUN: %libomptarget-compile-run-and-check-powerpc64-ibm-linux-gnu
+// RUN: %libomptarget-compile-run-and-check-powerpc64le-ibm-linux-gnu
+// RUN: %libomptarget-compile-run-and-check-x86_64-pc-linux-gnu
+// RUN: %libomptarget-compile-run-and-check-nvptx64-nvidia-cuda
+
+#include <stdio.h>
+
+// OpenMP 5.1, sec. 2.21.7.1 "map Clause", p. 351 L14-16:
+// "If the map clause appears on a target, target data, or target exit data
+// construct and a corresponding list item of the original list item is not
+// present in the device data environment on exit from the region then the
+// list item is ignored."
+
+int main(void) {
+  int f = 5, r = 6, d = 7, af = 8;
+
+  // Check exit from omp target data.
+  // CHECK: f = 5, af = 8
+  #pragma omp target data map(from: f) map(always, from: af)
+  {
+    #pragma omp target exit data map(delete: f, af)
+  }
+  printf("f = %d, af = %d\n", f, af);
+
+  // Check omp target exit data.
+  // CHECK: f = 5, r = 6, d = 7, af = 8
+  #pragma omp target exit data map(from: f) map(release: r) map(delete: d) \
+                               map(always, from: af)
+  printf("f = %d, r = %d, d = %d, af = %d\n", f, r, d, af);
+
+  return 0;
+}
diff --git a/openmp/libomptarget/test/mapping/present/target_exit_data.c b/openmp/libomptarget/test/mapping/present/target_exit_data_delete.c
rename from openmp/libomptarget/test/mapping/present/target_exit_data.c
rename to openmp/libomptarget/test/mapping/present/target_exit_data_delete.c
--- a/openmp/libomptarget/test/mapping/present/target_exit_data.c
+++ b/openmp/libomptarget/test/mapping/present/target_exit_data_delete.c
@@ -23,18 +23,18 @@
   fprintf(stderr, "addr=%p, size=%ld\n", &i, sizeof i);
 
   // CHECK-NOT: Libomptarget
-#pragma omp target enter data map(alloc: i)
-#pragma omp target exit data map(present, release: i)
+  #pragma omp target enter data map(alloc: i)
+  #pragma omp target exit data map(present, delete: i)
 
-  // CHECK: i is present
-  fprintf(stderr, "i is present\n");
+  // CHECK: i was present
+  fprintf(stderr, "i was present\n");
 
   // CHECK: Libomptarget message: device mapping required by 'present' map type modifier does not exist for host address 0x{{0*}}[[#HOST_ADDR]] ([[#SIZE]] bytes)
   // CHECK: Libomptarget fatal error 1: failure of target construct while offloading is mandatory
-#pragma omp target exit data map(present, release: i)
+  #pragma omp target exit data map(present, delete: i)
 
-  // CHECK-NOT: i is present
-  fprintf(stderr, "i is present\n");
+  // CHECK-NOT: i was present
+  fprintf(stderr, "i was present\n");
 
   return 0;
 }
diff --git a/openmp/libomptarget/test/mapping/present/target_exit_data.c b/openmp/libomptarget/test/mapping/present/target_exit_data_release.c
rename from openmp/libomptarget/test/mapping/present/target_exit_data.c
rename to openmp/libomptarget/test/mapping/present/target_exit_data_release.c
--- a/openmp/libomptarget/test/mapping/present/target_exit_data.c
+++ b/openmp/libomptarget/test/mapping/present/target_exit_data_release.c
@@ -23,18 +23,18 @@
   fprintf(stderr, "addr=%p, size=%ld\n", &i, sizeof i);
 
   // CHECK-NOT: Libomptarget
-#pragma omp target enter data map(alloc: i)
-#pragma omp target exit data map(present, release: i)
+  #pragma omp target enter data map(alloc: i)
+  #pragma omp target exit data map(present, release: i)
 
-  // CHECK: i is present
-  fprintf(stderr, "i is present\n");
+  // CHECK: i was present
+  fprintf(stderr, "i was present\n");
 
   // CHECK: Libomptarget message: device mapping required by 'present' map type modifier does not exist for host address 0x{{0*}}[[#HOST_ADDR]] ([[#SIZE]] bytes)
   // CHECK: Libomptarget fatal error 1: failure of target construct while offloading is mandatory
-#pragma omp target exit data map(present, release: i)
+  #pragma omp target exit data map(present, release: i)
 
-  // CHECK-NOT: i is present
-  fprintf(stderr, "i is present\n");
+  // CHECK-NOT: i was present
+  fprintf(stderr, "i was present\n");
 
   return 0;
 }
diff --git a/openmp/libomptarget/test/mapping/target_data_array_extension_at_exit.c b/openmp/libomptarget/test/mapping/target_data_array_extension_at_exit.c
--- a/openmp/libomptarget/test/mapping/target_data_array_extension_at_exit.c
+++ b/openmp/libomptarget/test/mapping/target_data_array_extension_at_exit.c
@@ -19,8 +19,8 @@
 
 // RUN: %libomptarget-compile-x86_64-pc-linux-gnu \
 // RUN:   -fopenmp-version=51 -DEXTENDS=BEFORE
-// XUN: %libomptarget-run-x86_64-pc-linux-gnu 2>&1 \
-// XUN: | %fcheck-x86_64-pc-linux-gnu
+// RUN: %libomptarget-run-x86_64-pc-linux-gnu 2>&1 \
+// RUN: | %fcheck-x86_64-pc-linux-gnu
 
 // --------------------------------------------------
 // Check extends after